Does a dog have the Buddha nature?
Sexual orientation
Your question reads, "...the context is a website whereby I allow the user to enter a sexuality." Part of your confusion starts here: using precise language to define your problem will help you find a solution. "Sexuality" is not precise enough. On your website, you are allowing users to enter their sexual orientation: that is the precise term you need.
By being more precise, we see that the information that you are allowing the users to enter is emphatically not the labels that the users apply to themselves (such as homosexual, gay, or lesbian). Rather, you are allowing the users to enter the gender of people to whom the users are sexually oriented.
What does LGBT mean?
Your question makes a passing reference to LGBT but only mentions "Gay and Lesbian." "LGBT is an initialism that stands for lesbian, gay, bisexual, and transgender." (Wikipedia, emphasis added.) People who label themselves as gay or lesbian almost always recognize at least a few other labels with equal validity to gay or lesbian. The extremely-common initialism, LGBT, demonstrates that in addition to gay and lesbian, many people describe their sexual orientation as bisexual or some other label.
Therefore, your basic question is a false dichotomy: you do not need to choose between “Homosexual” or “Gay and Lesbian” because there are clearly many other options than those three terms. Furthermore, "Homosexual" and presumably "Heterosexual" as the two choices for your users do not include people who label themselves as bisexual, so "Homosexual" is not broad enough. Your proposed alternative of "Gay and Lesbian" presumably presented with a choice such as "Straight" is also under-inclusive.
For your website, neither of the options you have presented are adequate.
What is gender?
The issue, however, is even more complicated than attempting to create a fully-inclusive list of labels for sexual orientation. Transgender (the "T" in LGBT) is not a sexual orientation: it describes a person's gender identity. The overwhelming majority of people only think of gender as male or female. This black-and-white perspective of gender is often called something like a "binary gender" model.
Some people, however, do not cleanly fit into these two gender definitions. Easy, non-controversial examples that defy binary-gender labels: some people have male and female sexual organs; some people are genetically female (two X-chromosomes), but have the secondary sexual characteristics of a male (sexual organs, for example); and some people have two X-chromosomes and a Y-chromosome, which, genetically, is not exactly male nor female, and the person could develop the secondary sexual characteristics of a male, a female, or an intersex person. When these people are presented with two options for gender, male or female, which should they choose and why?
Which brings us back to people who identify as transgender. Is their gender male or female or something else?
Your question is explicitly about sexual orientation, so we do not need to answer most of the difficult questions within gender identity, but it is related to sexual orientation, so when choosing the language for your website, you must be aware of the basic issues of gender identity.
How to allow your users to enter their sexual orientation
If your goal is to allow your users to describe their sexual orientation exactly as each user would like to describe their sexual orientation, then the best option is to write "Sexual orientation:" and then have a text box that allows users to type anything they want to type.
Of course, if you want to sort data based on sexual orientation, you need standardized data. Allowing everyone to enter free-text will give you problems such as three people who all use the same label to describe their sexual orientation, but they typed it differently: bisexual, bi-sexual, and bi sexual. Computers are stupid, and the computer will see those three descriptions as different rather than as the same.
Therefore, you likely will want to give your users a set of options from which to choose. If we start with the belief that there are only two genders, male and female, then your best design would likely be to write, "Sexual orientation:" then have two checkboxes, male and female. Each user could choose male, female, or both.
For the overwhelming majority of websites, the above method is satisfactory, not perfect, but satisfactory. Because it assumes binary gender, it is under-inclusive, but because most people who make websites have not wrestled with the difficult and contentious issues of gender identity, for most people, trying to create standardized options other than male and female would require tremendous effort for relatively little return.
As described above, the binary gender model, however, is not particularly good. In a perfect world, the English language would have a more full vocabulary to describe gender and most people would be familiar with the options other than male and female. We do not live in that world.
The solution to the errors inherent in binary-gender descriptions and its impact on describing sexual orientation, depend on the reasons a website wants to collect sexual orientation information about its users. An easy option that is 100% inclusive of all sexual orientations is to have three check boxes: male, female, and other. "Other" is terribly imprecise, but it is more inclusive.
Without knowing why you want sexual orientation and how you will use the data, it is impossible for anyone here to give you a specific list of choices for your users.
Conclusion
- Neither of your proposed options are appropriate.
- It is essential that you clearly understand that you are asking your
users for their sexual orientation and you must clearly understand
the range of possibilities within sexual orientation.
- Gender identity cannot be separated from sexual orientation, so you
will need to accept that the binary-gender model is not accurate and develop
some understanding of the complexity of gender identity.
- Do not ask your users to label their sexual orientation. Labels
include homosexual, gay, or lesbian.
- Do allow your users to enter the gender (or gender identity) of the
persons to whom the users are sexually oriented.